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The maximum maximum of a martingale with given n marginals

  • Pierre Henry-Labordere
  • Jan Obloj
  • Peter Spoida
  • Nizar Touzi
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    We obtain bounds on the distribution of the maximum of a continuous martingale with fixed marginals at finitely many intermediate times. The bounds are sharp and attained by a solution to n-marginal Skorokhod embedding problem in Obloj and Spoida (2013). It follows that their embedding maximises the maximum among all other embeddings. Our motivating problem is superhedging lookback options under volatility uncertainty for an investor allowed to dynamically trade the underlying asset and statically trade European call options for all possible strikes and finitely-many maturities. We derive a pathwise inequality which induces the cheapest superhedging value, which extends the two-marginals pathwise inequality of Brown, Hobson and Rogers (1998). This inequality, proved by elementary arguments, is obtained by following the stochastic control approach of Galichon, Henry-Labordere and Touzi (2011).

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    File URL: http://arxiv.org/pdf/1203.6877
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    Paper provided by arXiv.org in its series Papers with number 1203.6877.

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    Date of creation: Mar 2012
    Date of revision: Sep 2014
    Handle: RePEc:arx:papers:1203.6877
    Contact details of provider: Web page: http://arxiv.org/

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    1. Alexander Cox & Jan Obłój, 2011. "Robust pricing and hedging of double no-touch options," Finance and Stochastics, Springer, vol. 15(3), pages 573-605, September.
    2. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
    3. Mathias Beiglb\"ock & Pierre Henry-Labord\`ere & Friedrich Penkner, 2011. "Model-independent Bounds for Option Prices: A Mass Transport Approach," Papers 1106.5929, arXiv.org, revised Feb 2013.
    4. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-51, October.
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